Joseph Shoenfield writes:
> By a strictly pi-0-n+1 sentence I mean one which is not pi-0-n or
> sigma-0-n.
I still don't understand. By adding dummy quantifiers, it is trivial
to convert a Pi^0_1 sentence to a logically equivalent Pi^0_{n+1}
sentence which is not Pi^0_n or Sigma^0_n.
You could avoid this difficulty by talking about sentences up to
equivalence over ZFC or something of the sort, but I still don't see
the relevance of this.
> I stated this conjecture only because you demanded one,
> but I would really rather return to my original general statement:
OK, let's drop the discussion of your conjecture, which makes no sense
to me anyway.
> one should look for a result which relates the position of an
> undecidable statement in the arithmetical or analytic hierarchy and
> the number and kind of large cardinals needed to prove it.
Why? I don't see that such a relationship would have a bearing on any
important f.o.m. issue. It seems *much* more important to find good
examples of finite combinatorial statements that require large
cardinals to prove them.
> I have always been puzzled as to why you considered the particular
> result of Harvey such a key result in the completeness program.
The reason I view Harvey's independence result as key is that it is
the state of the art vis a vis the program that I mentioned, i.e. to
extend the incompleteness phenomenon into finite combinatorics, or
more specifically, to find finite combinatorial statements which are
independent of ZFC, or ZFC plus large cardinals. By state of the art
I mean the best result that is known at the present time.
> I find your challenge to explain why I value the Steel-Martin
> theorem so highly not only reasonable but welcome, since it will
> afford me a chance to express some thoughts on judging mathematical
> results which I have had recently. I hope to meet the challenge
> sometime soon.
I'm looking forward to it. I wonder how you are going to express your
thoughts on these matters without using any informal concepts.
-- Steve